# An uncertainty principle for unimodular quantum groups

## Abstract

We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.

- Authors:

- School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6 (Canada)
- (France)

- Publication Date:

- OSTI Identifier:
- 22306077

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; ENTROPY; GRAPH THEORY; QUANTUM GROUPS; RANDOMNESS; UNCERTAINTY PRINCIPLE

### Citation Formats

```
Crann, Jason, Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d'Ascq Cédex, and Kalantar, Mehrdad, E-mail: jason-crann@carleton.ca, E-mail: mkalanta@math.carleton.ca.
```*An uncertainty principle for unimodular quantum groups*. United States: N. p., 2014.
Web. doi:10.1063/1.4890288.

```
Crann, Jason, Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d'Ascq Cédex, & Kalantar, Mehrdad, E-mail: jason-crann@carleton.ca, E-mail: mkalanta@math.carleton.ca.
```*An uncertainty principle for unimodular quantum groups*. United States. doi:10.1063/1.4890288.

```
Crann, Jason, Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d'Ascq Cédex, and Kalantar, Mehrdad, E-mail: jason-crann@carleton.ca, E-mail: mkalanta@math.carleton.ca. Fri .
"An uncertainty principle for unimodular quantum groups". United States.
doi:10.1063/1.4890288.
```

```
@article{osti_22306077,
```

title = {An uncertainty principle for unimodular quantum groups},

author = {Crann, Jason and Université Lille 1 - Sciences et Technologies, UFR de Mathématiques, Laboratoire de Mathématiques Paul Painlevé - UMR CNRS 8524, 59655 Villeneuve d'Ascq Cédex and Kalantar, Mehrdad, E-mail: jason-crann@carleton.ca, E-mail: mkalanta@math.carleton.ca},

abstractNote = {We present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to q-traces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.},

doi = {10.1063/1.4890288},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 15 00:00:00 EDT 2014},

month = {Fri Aug 15 00:00:00 EDT 2014}

}